Pour heavy cream and whole milk into a medium saucepan and heat to 150 degrees Fahrenheit.
Add in 1-2 whole cups of mint leaves, give a light stir, then cover the saucepan with the lid and let the mixture steep for 2 hours.
Strain out the leaves and pour the mixture back into the saucepan. Begin heating to 150F once again.
Meanwhile, add sugar, cornstarch, and salt to a large, heatproof glass bowl and stir with a whisk to combine.
Once the milk mixture is heated, pour a splash of it into the sugar bowl and stir until smooth. Make sure no lumps of cornstarch are left behind.
Keep adding splashes of cream, pausing to stir in between, until there's about half of the cream left in the saucepan. The goal is to help the sugar/cornstarch mixture assimilate with the rest of the cream.
Pour the now smooth and thinned sugar and cream mixture into the saucepan with the rest of it. Turn the heat to medium low.
Stir continuously with a whisk until the mixture starts to gently rise and thicken. This should take anywhere from 3-5 minutes. Do not let it boil or the cream will curdle! :( The mixture is done when it resembles a thin pudding and a tail begins to form behind the whisk.
Immediately remove from heat and pour into a large glass heatproof bowl. Add 1/4 tsp of peppermint extract and stir to combine.
Cover the bowl with plastic wrap, making sure to touch the cream with the plastic. This will prevent a skin from forming.
Place in the fridge for 8 hours or overnight until the cream is completely cold. (The longer it sits, the deeper the flavors will meld, and the creamier your ice cream will be.)
Once ready to churn, pour ice cream into your ice cream maker. Set it to 'stir' if using a Kitchenaid stand mixer and let it go for about 10 minutes.
Meanwhile, add 2 tsp of oil to your chocolate shavings and melt in a glass bowl over a double-boiler. Stir slowly and consistently until completely melted. Set aside.
The cream is done when it looks thick with very large lumps swirling in the bowl. If using a kitchenaid mixer, it will be touching the attachment holding the churning device and have ice cream circling around the outer edges. Stop here.
Dip a spatula into your chocolate and drizzle over the ice cream, stir using a separate spatula, then repeat the process until you've used all of the chocolate. The ice cream should now have plenty of beautiful chocolate shavings scattered throughout.
Scoop into a 1 or 1.5qt airtight container and place in the freezer. Depending on what size you've chosen you may have some ice cream leftover. Enjoy!